Problem 28 - Precalculus Essentials

Question

$$\cos \theta=\frac{1}{3}, \quad 270^{\circ}<\theta<360^{\circ}$$

Step-by-Step Solution

Answer

Based on the solution steps, \(\theta = 289.47°\) is found which falls within the given range \( 270° < \theta < 360° \).

Step 1: Find the reference angle

The reference angle, also known as the acute angle, can be found using the formula \( \cos^{-1}\left(\frac{1}{3}\right) \). This gives the reference angle in the first quadrant, which is about \( 70.53° \).

Step 2: Calculate the angle in fourth quadrant

Given the range, \( \theta \) must lie in the fourth quadrant. In the fourth quadrant, the cos function is positive. We have to subtract the reference angle from 360° since the terminal side is \( 360° - \text{reference angle} \). So, \( \theta = 360° - 70.53° = 289.47° \). This value falls within the specified range of \( 270° < \theta < 360° \) and is therefore the angle we are searching for.

Key Concepts

Reference Angle

A reference angle in trigonometry is an acute angle that helps us determine the properties of an original angle. Understanding this can be crucial for solving trigonometric problems.

The reference angle is always between 0° and 90°, and it is the smallest angle that the terminal side of an angle makes with the x-axis. This can simplify calculations, especially because trigonometric functions have symmetry across quadrants.

• To find the reference angle, you use the inverse trigonometric function on the given value. For example, when given a cosine value, calculate the reference angle by finding the angle in the first quadrant: \[\theta = \cos^{-1}\left(\frac{1}{3}\right) \approx 70.53°.\]
• The reference angle is useful because it helps us understand how the angle relates to the x-axis, regardless of its original quadrant.

Fourth Quadrant

The fourth quadrant in the coordinate plane is defined by angles ranging from 270° to 360°. This quadrant holds special character because of the behavior of trigonometric functions there.

In the fourth quadrant:

• The cosine and secant functions are positive, while sine, cosecant, tangent, and cotangent functions are negative.
• Knowing this helps you predict the sign of a trigonometric function result quickly without calculation.
• When finding an angle in this quadrant, one common way is to subtract the reference angle from 360°, ensuring that the calculated angle fits the quadrant's range.

For instance, with a reference angle of about 70.53°, the angle in the fourth quadrant is:\[\theta = 360° - 70.53° = 289.47°.\]

Cosine Function

The cosine function represents a key trigonometric function, used to understand the relationship between an angle and the lengths of the sides of a right triangle.

Cosine particularly measures the ratio of the adjacent side to the hypotenuse in a right triangle. It also describes various features of the unit circle.

• In the unit circle, the cosine function corresponds to the x-coordinate of a point at a given angle from the positive x-axis.
• Cosine values vary between -1 and 1. Positive values indicate the angle lies in the first or fourth quadrants, while negatives indicate the second or third quadrants.
• For an angle where \(\cos \theta = \frac{1}{3}\), it confirms that the angle lies in one of the quadrants where cosine is positive, narrowing it down to either the first or fourth quadrants.

Understanding these properties of the cosine function enables easier navigation and prediction when dealing with angles across different quadrants.